Volume 39, Issue 1 - Special Issue Dedicated to the 80th Anniversary of Professor Nikolay Yanev a.k.a. The Captain

This Special Issue is dedicated to the 80th anniversary of Nikolay Yanev, a renowned Bulgarian mathematician and an eminent member of the international branching processes community. We present a brief description of his academic career path and his role as a founder of the Bulgarian school of branching processes and their applications.

The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

We consider a special class of Galton–Watson theta-processes in a varying environment fully defined by four parameters, with two of them ( θ , r ) {(\theta,r)} being fixed over time n , and the other two ( a n , c n ) {(a_{n},c_{n})} characterizing the altering reproduction laws. We establish a sequence of transparent limit theorems for the theta-processes with possibly defective reproduction laws. These results may serve as a stepping stone towards incisive general results for the Galton–Watson processes in a varying environment.

This research focuses on stochastic modeling of the evolution over time of biological populations through branching processes. We introduce a new class of discrete time two-sex branching processes with random mating and overlapping generations. Mating and reproduction are considered to be influenced by the numbers of females and males existing in the population. This evolution over time with generational overlap is a novel research in two-sex branching process literature. We study general probabilistic properties and establish some limiting results of biological interest.

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

We give a survey of some recent results describing some properties of branching processes under nonstandard conditions.